There will be two arrays say y1(nx1) and y2(nx1) for n space grids.. and the equations are of the form d/dt(y1 or y2) = F1 or F2 (yi,y'i,y"i...) , i=1,2. then you get two error arrays (nx1) for y1 and y2...
now how to use these to modify dt (time step) for this process ? modifying dt for each space point and for both the variables is very hard.. as you have to calculate them simultaneously..
Can I take
a = MAX[error_y1(nx1)]
b = MAX[error_y2(nx1)]
then compare a and b and modify dt according to the max error in the process. It looks reasonable
but I am not sure. Anyone encountered this same problem? Please help.
just go for any numerical methods book (good one) you will find pde using RK
Deb,
the statement "modifying dt for each space point and for both the variables is very hard", is not clear to me... If your PDE are non-linear I suggest to use a different integrator, in general i've never used rk for non linear pde (es cahn-allen, cahn hilliard), tough I've ever used semi implicit methods. But if you have to use rk the criteria of maximum variation over the fieldS is correct
Regards,
Fabio
Thank you Fabio, I am not sure that this rkf45 is the best method or not, myself....
Let me do some more tests... I will ask for help again :) .
NB: "modifying dt for each space point and for both the variables is very hard"
by this I meant that when you break up a PDE in finite differentiation in many ODE s, then time evolution of these ODE s with different time steps for different ODE s when they are related with spatial derivatives, sounds very difficult to me. Don't know if I could clear your confusion.
Regards,
Deb.
Dear Banerjee,
To use different time steps in each spatial locations you should search a little about "asynchronous time stepping" (or "multiple time steps", the exact terms vary somewhat).
The expression that control the adaptation of RKF has been developed for a single variable. In the optimal step calculation you will be asked to divide for a "difference of predicted values", which must be appropriatedly defined for a vector (1-norm?).
If one of the variables turns very stiff, the smaller time step will become very small (maybe order of magnitudes smaller) compared to the others, and that is where the need for asynchronous time stepping came from.
The use of non-local time stepping schemes, like semi-implicit, will require many anterior points if you try to go high-order, with non-constant time lags, which will demand some effort to generate the coefficients. I don't know if this would cancel your time savings, but it is point of attention.
Best regards
@ Mr. Bortolon,
Thank you, I did not know about asynchronous time stepping, but I think I should
find a simpler way to get the numerics done.. looks too complicated for me and not even sure that it will solve my problem.
Did not finally use RKF45 for coupled pdes but
attaching a simple ODE example of Runge-Kutta-Fehlberg method. May be useful to someone.
(written in Fortran, tested in ubuntu. added a script + example results)
Regards.
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